Modular invariant partition function of critical dense polymers

Morin-Duchesne, Alexi, Pearce, Paul A. and Rasmussen, Jørgen (2013) Modular invariant partition function of critical dense polymers. Nuclear Physics, Section B, 874 1: 312-357. doi:10.1016/j.nuclphysb.2013.05.016

Author Morin-Duchesne, Alexi
Pearce, Paul A.
Rasmussen, Jørgen
Title Modular invariant partition function of critical dense polymers
Journal name Nuclear Physics, Section B   Check publisher's open access policy
ISSN 0550-3213
Publication date 2013-09-01
Sub-type Article (original research)
DOI 10.1016/j.nuclphysb.2013.05.016
Open Access Status DOI
Volume 874
Issue 1
Start page 312
End page 357
Total pages 46
Place of publication Netherlands
Publisher Elsevier North-Holland
Collection year 2014
Language eng
Formatted abstract
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin–Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley–Lieb algebra extended by the shift operators 𝛺±1. In this enlarged algebra, the non-contractible loop fugacity is 𝛼 and the contractible loop fugacity is 𝛽. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley–Lieb algebra with a parameter 𝑣 that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on 𝑣. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Tr𝑑 (with a fixed number of defects 𝑑) and Chebyshev polynomials of the first kind. For critical dense polymers, where 𝛽=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on 𝑑 and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to 𝛼=2 in which case the traces are evaluated 𝑣=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions.
Keyword Conformal field-theory
Logarithmic operators
2 dimensions
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2014 Collection
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Citation counts: TR Web of Science Citation Count  Cited 6 times in Thomson Reuters Web of Science Article | Citations
Scopus Citation Count Cited 5 times in Scopus Article | Citations
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